A mechanical system's power output G(t)G(t)G(t) at time ttt is modeled for t>3t > 3t>3 by the function
G(t)=t4−t3−4t2+3t−2t2−t−6 G(t) = \frac{t^4 - t^3 - 4t^2 + 3t - 2}{t^2 - t - 6} G(t)=t2−t−6t4−t3−4t2+3t−2Given that
G(t)≡t2+A+Bt−3 G(t) \equiv t^2 + A + \frac{B}{t - 3} G(t)≡t2+A+t−3Bfind the value of the constant AAA and show that B=5B = 5B=5.
The curve KKK has equation y=G(t)y = G(t)y=G(t) for t>3t > 3t>3.
Find the equation of the tangent to KKK at the point where t=4t = 4t=4. Give your answer in the form y=mt+cy = mt + cy=mt+c, where mmm and ccc are constants to be found.
The region SSS is bounded by the curve KKK, the ttt-axis, and the vertical lines with equations t=4t = 4t=4 and t=5t = 5t=5. Find the exact area of SSS, writing your answer in the form a+bln2a + b \ln 2a+bln2, where aaa and bbb are constants to be found.
